Here are some very popular songs that were written in the 1920s or 1930s and have survived very well:

Here are some songs that were written in the '20s/'30s which have "survived" over the decades:

5'2 (Has Anybody Seen My Gal?)
Ain't Misbehavin'
Ain't She Sweet?
All Of Me
Always
Any Time
Are You Lonesome Tonight?
As Time Goes By
Baby Face
Basin Street Blues (may be older)
Beer Barrel Polka
Blue Hawaii
Blue Moon
Blue Skies
Brother, Can You Spare A Dime?
Button Up Your Overcoat
Bye-bye, Blackbird
Bye-bye, Blues
Dream A Little Dream Of Me
Georgia
Harbor Lights
Have You Ever Been Lonely?
Heart And Soul
Heart Of My Heart
I'll Get By
I'm Lookin' Over A Four-Leaf Clover
I'm Sitting On Top Of The World
If You Knew Susie
It Ain't Necessarily So
It Had To Be You
It's A Sin To Tell A Lie
It's Only a Paper Moon
Jeepers, Creepers!
Just Because
Lazy Bones
Lovesick Blues
Makin' Whoopee
Marie
Me And My Shadow
My Blue Heaven
My Buddy
My Funny Valentine
My Happiness
Nice Work If You Can Get It
Our Love Is Here To Stay
Over The Rainbow
Paper Doll
Red Sails In The Sunset
San Antonio Rose
Second-hand Rose
September In The Rain
September Song
Shanty In Old Shanty Town
Side By Side
Someone To Watch Over Me
Sonny Boy
Stardust
Stormy Weather
Summertime
Swanee
Tea For Two
The Glory Of Love
Them There Eyes
They Can't Take That Away From Me
Three Little Fishes
Up A Lazy River